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Geometric series proof [Edexcel C2]

10. A geometric series is

a + ar + ar^2 + ...

a) Prove that the sum of the first

n

terms of this series is given by

S_n = \frac{a(1-r^n)}{1-r}

[4 marks]

Obviously that's the formula they give you so I've always sort of taken it as a given. Thus I have no idea how to prove that it is true...can anyone help me out and guide me through how to answer this question?

Thanks. :smile:

Reply 1

Notnek

s_n=a+ar+ar^2+...

What's

rs_n

and then what's

s_n-rs_n

?

That should help you prove it.

Reply 2

IChem

S_n = a+ ar +ar^2........ar^{n-2} + ar^{n-1}

rS_n = ar+ ar^2 +ar^3........ar^{n-1} + ar^n

S_n-rS_n =a-ar^n

S_n(1-r) = a(1-r^n)

S_n= \frac{a(1-r^n)}{1-r}

Geddit?

Thanks!

Although 3 years is a while for a thanks

Reply 5

johnandjennyjohn

I looked at the answer and used a quotient of (1 - r) top and bottom which cancels a lot of terms on top. Not formal but it's nice!

Reply 6

plxntsoul

Original post by IChem

S_n = a+ ar +ar^2........ar^{n-2} + ar^{n-1}

rS_n = ar+ ar^2 +ar^3........ar^{n-1} + ar^n

S_n-rS_n =a-ar^n

S_n(1-r) = a(1-r^n)

S_n= \frac{a(1-r^n)}{1-r}

Geddit?

7 years later and still useful :wink:

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